Subgroups — C₂×C₅×C₃⋊D₃×ΓL(1,𝔽₃₂) #42124830

Order 27900.

480 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₅×ΓL(1,𝔽₃₂)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×C₃⋊D₃×ΓL(1,𝔽₃₂)	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₅×D₃×ΓL(1,𝔽₃₂)	—	3	4	no	maximal
4	C₃²×C₅×ΓL(1,𝔽₃₂)	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×C₅×C₃⋊D₃×C₃₁	C₅	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃⋊D₃×ΓL(1,𝔽₃₂)	C₅	1	5	yes	maximal, direct product factor
7	C₂×C₃×C₅×ΓL(1,𝔽₃₂)	D₃	1	4	yes	complemented
8	C₅×D₃×ΓL(1,𝔽₃₂)	—	3	8	no	—
9	C₂²×C₅×ΓL(1,𝔽₃₂)	—	9	1	no	Hall
10	C₅×C₃⋊D₃×C₃₁	C₂×C₅	1	2	yes	complemented
11	C₂×C₃²×ΓL(1,𝔽₃₂)	C₂×C₅	1	5	yes	complemented
12	C₃⋊D₃×ΓL(1,𝔽₃₂)	C₂×C₅	1	10	yes	direct product factor
13	C₂×C₃²×C₅×C₃₁	C₂×C₅	1	1	yes	characteristic, complemented
14	C₃×C₅×ΓL(1,𝔽₃₂)	C₂×D₃	1	4	yes	complemented
15	C₂×C₅×D₃×C₃₁	—	3	4	no	—
16	C₂×D₃×ΓL(1,𝔽₃₂)	—	3	20	no	—
17	C₂×C₅×ΓL(1,𝔽₃₂)	—	9	2	no	—
18	C₂×C₅×ΓL(1,𝔽₃₂)	C₃⋊D₃	1	1	yes	characteristic, direct product factor
19	C₃²×C₅×C₃₁	C₂²×C₅	1	1	yes	characteristic, complemented
20	C₃²×ΓL(1,𝔽₃₂)	C₂²×C₅	1	5	yes	complemented
21	C₂×C₃⋊D₃×C₃₁	C₅²	1	1	yes	characteristic, complemented, Hall
22	C₂×C₃×C₅×C₃₁	C₅×D₃	1	4	yes	complemented
23	C₅×D₃×C₃₁	—	3	8	no	—
24	C₂×C₃×ΓL(1,𝔽₃₂)	C₅×D₃	1	20	yes	complemented
25	D₃×ΓL(1,𝔽₃₂)	—	3	40	no	—
26	C₂×C₅²×C₃⋊D₃	—	31	1	no	maximal, Hall
27	C₅×ΓL(1,𝔽₃₂)	C₂×C₃⋊D₃	1	1	yes	characteristic, direct product factor, Hall
28	C₂²×C₅×C₃₁	—	9	1	no	—
29	C₂²×ΓL(1,𝔽₃₂)	—	9	5	no	—
30	C₂×C₃²×C₃₁	C₂×C₅²	1	1	yes	characteristic, complemented
31	C₃⋊D₃×C₃₁	C₂×C₅²	1	2	yes	complemented
32	C₃×C₅×C₃₁	C₂×C₅×D₃	1	4	yes	complemented
33	C₃×ΓL(1,𝔽₃₂)	C₂×C₅×D₃	1	20	yes	complemented
34	C₂×C₃²×C₅²	—	31	1	no	—
35	C₅²×C₃⋊D₃	—	31	2	no	—
36	C₂×D₃×C₃₁	—	3	4	no	—
37	C₂×C₅×C₃₁	—	9	2	no	—
38	C₂×C₅×C₃₁	C₅×C₃⋊D₃	1	1	yes	characteristic, complemented
39	C₂×ΓL(1,𝔽₃₂)	—	9	10	no	—
40	C₂×ΓL(1,𝔽₃₂)	C₅×C₃⋊D₃	1	5	yes	direct product factor
41	C₂×C₅²×D₃	—	93	4	no	—
42	C₃²×C₃₁	C₂²×C₅²	1	1	yes	characteristic, complemented, Hall
43	C₃²×C₅²	—	31	1	no	Hall
44	C₂×C₃×C₃₁	C₅²×D₃	1	4	yes	complemented
45	D₃×C₃₁	—	3	8	no	—
46	C₂×C₅×C₃⋊D₃	—	31	5	no	—
47	C₂×C₅×C₃⋊D₃	ΓL(1,𝔽₃₂)	1	1	yes	characteristic, direct product factor
48	C₅×C₃₁	C₂×C₅×C₃⋊D₃	1	1	yes	characteristic, complemented
49	ΓL(1,𝔽₃₂)	C₂×C₅×C₃⋊D₃	1	5	yes	direct product factor
50	C₂×C₃×C₅²	—	31	4	no	—
51	C₅²×D₃	—	93	8	no	—
52	C₂²×C₃₁	—	9	1	no	Hall
53	C₂²×C₅²	—	279	1	no	Hall
54	C₃×C₃₁	C₂×C₅²×D₃	1	4	yes	complemented
55	C₂×C₃²×C₅	—	31	5	no	—
56	C₂×C₃²×C₅	C₂×ΓL(1,𝔽₃₂)	1	1	yes	characteristic, complemented
57	C₅×C₃⋊D₃	—	31	10	no	—
58	C₅×C₃⋊D₃	C₂×ΓL(1,𝔽₃₂)	1	2	yes	direct product factor
59	C₃×C₅²	—	31	4	no	—
60	C₂×C₃₁	—	9	2	no	—
61	C₂×C₃₁	C₅²×C₃⋊D₃	1	1	yes	characteristic, complemented
62	C₂×C₅×D₃	—	3	4	no	—
63	C₂×C₅×D₃	—	93	20	no	—
64	C₂×C₅²	—	279	2	no	—
65	C₂×C₅²	—	31	1	no	—
66	C₃²×C₅	—	31	5	no	—
67	C₃²×C₅	C₂²×ΓL(1,𝔽₃₂)	1	1	yes	characteristic, complemented
68	C₂×C₃⋊D₃	C₅×ΓL(1,𝔽₃₂)	1	1	yes	characteristic, direct product factor, Hall
69	C₃₁	C₂×C₅²×C₃⋊D₃	1	1	yes	characteristic, complemented, Sylow
70	C₂×C₃×C₅	—	31	20	no	—
71	C₂×C₃×C₅	D₃×ΓL(1,𝔽₃₂)	1	4	yes	complemented
72	C₅×D₃	—	3	8	no	—
73	C₅×D₃	—	93	40	no	—
74	C₅²	—	31	1	no	Sylow
75	C₂²×C₅	—	9	1	no	—
76	C₂²×C₅	—	279	5	no	—
77	C₂×C₃²	C₂×C₅×ΓL(1,𝔽₃₂)	1	1	yes	characteristic, complemented
78	C₃⋊D₃	C₂×C₅×ΓL(1,𝔽₃₂)	1	2	yes	direct product factor
79	C₃×C₅	—	31	20	no	—
80	C₃×C₅	C₂×D₃×ΓL(1,𝔽₃₂)	1	4	yes	complemented
81	C₂×D₃	—	3	4	no	—
82	C₂×C₅	—	9	2	no	—
83	C₂×C₅	—	279	10	no	—
84	C₂×C₅	—	31	5	no	—
85	C₂×C₅	C₃⋊D₃×ΓL(1,𝔽₃₂)	1	1	yes	characteristic, direct product factor
86	C₃²	C₂²×C₅×ΓL(1,𝔽₃₂)	1	1	yes	characteristic, complemented, Sylow
87	C₂×C₃	C₅×D₃×ΓL(1,𝔽₃₂)	1	4	yes	complemented
88	D₃	—	3	8	no	—
89	C₅	—	31	5	no	—
90	C₅	C₂×C₃⋊D₃×ΓL(1,𝔽₃₂)	1	1	yes	characteristic, direct product factor
91	C₂²	—	9	1	no	Sylow
92	C₃	C₂×C₅×D₃×ΓL(1,𝔽₃₂)	1	4	yes	complemented
93	C₂	—	9	2	no	—
94	C₂	C₅×C₃⋊D₃×ΓL(1,𝔽₃₂)	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₅	C₃⋊D₃×ΓL(1,𝔽₃₂)
commutator, cosocle, hypocenter	C₃²×C₃₁	C₂²×C₅²
Frattini	C₁	G
Fitting, socle	C₂×C₃²×C₅×C₃₁	C₂×C₅
LCS	G ▷ C₃²×C₃₁
UCS	C₁ ◁ C₂×C₅

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×C₃²×C₅×ΓL(1,𝔽₃₂)	C₂	semidirect
2. C₅×C₃⋊D₃×ΓL(1,𝔽₃₂)	C₂	direct (N × H)
4. C₃²×C₅×ΓL(1,𝔽₃₂)	C₂²	semidirect
5. C₂×C₅×C₃⋊D₃×C₃₁	C₅	semidirect
6. C₂×C₃⋊D₃×ΓL(1,𝔽₃₂)	C₅	direct (N × H)
7. C₂×C₃×C₅×ΓL(1,𝔽₃₂)	D₃	semidirect
10. C₅×C₃⋊D₃×C₃₁	C₂×C₅	semidirect
11. C₂×C₃²×ΓL(1,𝔽₃₂)	C₂×C₅	semidirect
12. C₃⋊D₃×ΓL(1,𝔽₃₂)	C₂×C₅	direct (N × H)
13. C₂×C₃²×C₅×C₃₁	C₂×C₅	semidirect
14. C₃×C₅×ΓL(1,𝔽₃₂)	C₂×D₃	semidirect
18. C₂×C₅×ΓL(1,𝔽₃₂)	C₃⋊D₃	direct (N × H)
19. C₃²×C₅×C₃₁	C₂²×C₅	semidirect
20. C₃²×ΓL(1,𝔽₃₂)	C₂²×C₅	semidirect
21. C₂×C₃⋊D₃×C₃₁	C₅²	semidirect
22. C₂×C₃×C₅×C₃₁	C₅×D₃	semidirect
24. C₂×C₃×ΓL(1,𝔽₃₂)	C₅×D₃	semidirect
27. C₅×ΓL(1,𝔽₃₂)	C₂×C₃⋊D₃	direct (N × H)
30. C₂×C₃²×C₃₁	C₂×C₅²	semidirect
31. C₃⋊D₃×C₃₁	C₂×C₅²	semidirect
32. C₃×C₅×C₃₁	C₂×C₅×D₃	semidirect
33. C₃×ΓL(1,𝔽₃₂)	C₂×C₅×D₃	semidirect
38. C₂×C₅×C₃₁	C₅×C₃⋊D₃	semidirect
40. C₂×ΓL(1,𝔽₃₂)	C₅×C₃⋊D₃	direct (N × H)
42. C₃²×C₃₁	C₂²×C₅²	semidirect
44. C₂×C₃×C₃₁	C₅²×D₃	semidirect
47. C₂×C₅×C₃⋊D₃	ΓL(1,𝔽₃₂)	direct (N × H)
48. C₅×C₃₁	C₂×C₅×C₃⋊D₃	semidirect
49. ΓL(1,𝔽₃₂)	C₂×C₅×C₃⋊D₃	direct (N × H)
54. C₃×C₃₁	C₂×C₅²×D₃	semidirect
56. C₂×C₃²×C₅	C₂×ΓL(1,𝔽₃₂)	semidirect
58. C₅×C₃⋊D₃	C₂×ΓL(1,𝔽₃₂)	direct (N × H)
61. C₂×C₃₁	C₅²×C₃⋊D₃	semidirect
67. C₃²×C₅	C₂²×ΓL(1,𝔽₃₂)	semidirect
68. C₂×C₃⋊D₃	C₅×ΓL(1,𝔽₃₂)	direct (N × H)
69. C₃₁	C₂×C₅²×C₃⋊D₃	semidirect
71. C₂×C₃×C₅	D₃×ΓL(1,𝔽₃₂)	semidirect
77. C₂×C₃²	C₂×C₅×ΓL(1,𝔽₃₂)	semidirect
78. C₃⋊D₃	C₂×C₅×ΓL(1,𝔽₃₂)	direct (N × H)
80. C₃×C₅	C₂×D₃×ΓL(1,𝔽₃₂)	semidirect
85. C₂×C₅	C₃⋊D₃×ΓL(1,𝔽₃₂)	direct (N × H)
86. C₃²	C₂²×C₅×ΓL(1,𝔽₃₂)	semidirect
87. C₂×C₃	C₅×D₃×ΓL(1,𝔽₃₂)	semidirect
90. C₅	C₂×C₃⋊D₃×ΓL(1,𝔽₃₂)	direct (N × H)
92. C₃	C₂×C₅×D₃×ΓL(1,𝔽₃₂)	semidirect
94. C₂	C₅×C₃⋊D₃×ΓL(1,𝔽₃₂)	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃²×C₅×ΓL(1,𝔽₃₂)	2
C₅×C₃⋊D₃×ΓL(1,𝔽₃₂)	2
C₂×C₅×D₃×ΓL(1,𝔽₃₂)	3
C₂×C₅×C₃⋊D₃×C₃₁	5
C₂×C₃⋊D₃×ΓL(1,𝔽₃₂)	5
C₂×C₅²×C₃⋊D₃	31

Subgroups — C2×C5×C3⋊D3×ΓL(1,𝔽32) #42124830

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×C₃⋊D₃×ΓL(1,𝔽₃₂) #42124830